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For an entire function $f(z)=\sum_{n\geq0}a_nz^n$ write $\mu(r)=\max_n|a_n|r^n$ for its maximum term and $M(r)=\max_{|z|=r}|f(z)|$ for its maximum modulus. Problem 227 asks whether, for $f$ not a polynomial, the ratio $\mu(r)/M(r)$ can only tend to $0$ when it has a limit. Clunie and Hayman show that it can tend to any prescribed value: for every $\lambda\in[0,1/2]$ there is a transcendental entire function with

$$\lim_{r\to\infty}\frac{\mu(r)}{M(r)}=\lambda.$$

Any $\lambda>0$ is a counterexample, so the answer to the question is no. The site's Problem 513 asks a different question about the same ratio, the greatest possible value of its limit inferior, and is open. The paper is not held in the library, so its construction is not compiled here; this page records the result as the site credits it.

Acceptance. The paper is refereed: J. Clunie and W. K. Hayman, The maximum term of a power series, J. Analyse Math. 12 (1964), 143--186. The site's curator, Thomas F. Bloom, marks Problem 227 disproved and credits this paper with the disproof. The page is dated by the first day of the issue month, since the paper's first posting carries no finer date.